博碩士論文 100222017 詳細資訊




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姓名 蔡秉諺(Ping-yen Tsai)  查詢紙本館藏   畢業系所 物理學系
論文名稱 自旋粒子在萊斯納-諾思通黑洞的生成
(Spin Particle Creation in Reissner-Nordström Black Holes)
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摘要(中) 本論文以狄拉克方程式(Dirac equation)計算在帶電的近極端萊斯納-諾思通黑洞(Reissner-Nordström Black Holes)幾何背景時空下,事件視界(event horizon)附近的自旋場(spin field)自發性的粒子成對產生。由於量子擾動,空間中會隨機產生虛粒子對並快速湮滅,但有機會在湮滅前被視界抓住其中一個而使另一個散射出去,形成霍金輻射(Hawking radiation)。此效應等價於在加速的空間中,原本的量子真空態會觀察到額外的粒子,經由守恆流可計算粒子透射與反射通量的比率,藉此可得到射出粒子的產生率。

當黑洞所帶質量與電荷相等,稱為極端萊斯納-諾思通黑洞,而在近極端萊斯納-諾思通黑洞的近視界時空幾何,可看成二維的反德西特空間(AdS)鑲嵌球殼(sphere)。狄拉克方程式在這種彎曲空間中雖然需要加上額外的項,但是可以藉由將自旋場乘上特定函數讓方程式變回無額外項的微分方程。然後,藉由特定假設,將四元聯立偏微分方程轉為二元聯立微分方程,進而計算出解與其趨近無限與接近視界的守恆流。將趨近無限的向內走的流量設為零,而計算其他流量的比值即為本篇論文的目標。
摘要(英) This thesis uses the Dirac equation to calculate the spontaneous spin particle pair creation near event horizon of the near-extremal Reissner-Nordström black holes. In vacuum, virtual particles are randomly generated, due to the quantum perturbations, and quickly annihilated. There are possibilities that one of a particle pair can be caught into horizon before annihilation and the other is scattered out to infinity producing Hawking radiation. This effect is equivalent to the extra particle which is observed from a quantum vacuum by an accelerated observer. The ratio of the incoming/outgoing particle fluxes can be obtained by computing the transmitted and reflected conserved currents, then one can obtain the production rate of emitted particles.

A black hole which carries equal mass and charge is known as an extremal Reissner-Nordström black hole. The spacetime geometry of the near-horizon near-extremal Reissner-Nordström black hole is a two-dimensional anti-de Sitter(AdS) space cross a sphere. Dirac equation in this curved spacetime needs to add connection terms which can be removed by multiplying a certain function on the spin field. Then, by imposing a particular assumption, the four variables coupled partial differential equations can be simplified to two variables coupled differential equations. The exact solution can be obtained and the conserved currents at asymptotic and near horizon can be computed. By imposing no in-going current at asymptotic infinity, computing ratios of the other currents is the main goal of this thesis.
關鍵字(中) ★ RN黑洞
★ 帶電荷黑洞
★ 霍金輻射
★ 自旋場
★ 狄拉克方程
關鍵字(英) ★ RN black hole
★ charged black hole
★ Hawking radiation
★ spin field
★ Dirac equation
論文目次 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1
2 Quantum Field in Curved Spacetime . . . . . . . . . . . 3
2.1 Particle Production . . . . . . . . . . . . . . . . . 4
2.2 Charge Flux of Field. . . . . . . . . . . . . . . . . 6
2.3 Probing the Emission. . . . . . . . . . . . . . . . . 8
3 Charged Black Holes . . . . . . . . . . . . . . . . . . 9
3.1 Reissner-Nordström Geometric Structure . . . . . . . 10
3.2 Near Horizon Geometry for Near Extremal Black Hole . 11
3.3 Anti-de Sitter Space (AdS2)  Sphere (S2) . . . . . . 12
4 Spin Particle Creation . . . . . . . . . . . . . . . . 14
4.1 Dirac Equation in Flat Space-time. . . . . . . . . . 14
4.2 Dirac Equation in Curved Space-time. . . . . . . . . 15
4.3 Exact Solution of Spinor Field . . . . . . . . . . . 17
4.4 Production Rate. . . . . . . . . . . . . . . . . . . 19
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 22

A Spin Connection . . . . . . . . . . . . . . . . . . . . 24
B Solution of Spinor Field. . . . . . . . . . . . . . . . 26
C Asymptotic Limits and relations of Functions. . . . . . 30
C.1 Asymptotic Limits of Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . .30
C.2 Some Properties of Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . 30
Bibliography . . . . . . . . . . . . . . . . . . 31
參考文獻 [1] P. K. Townsend, "Black Holes: Lecture Notes," gr-qc/9707012.
[2] S. W. Hawking, "Particle Creation by Black Holes," Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)].
[3] J. D. Bekenstein, "Black Holes and Entropy," Phys. Rev. D 7, 2333 (1973).
[4] J. M. Bardeen, B. Carter and S. W. Hawking, "The Four Laws of Black Hole Mechanics," Commun. Math. Phys. 31, 161 (1973).
[5] M. K. Parikh and F. Wilczek, "Hawking Radiation as Tunneling," Phys. Rev. Lett. 85, 5042 (2000) [hep-th/9907001].
[6] S. P. Kim and D. N. Page, "Schwinger Pair Production in dS(2) and AdS(2),"
Phys. Rev. D 78 (2008) 103517 [arXiv:0803.2555 [hep-th]].
[7] I.-C. Lin, "Spontaneous Pair Production in Reissner-Nordstrom Black Holes", MSc. Thesis, (National Central University) (2012).
[8] C. -M. Chen, S. P. Kim, I-C. Lin, J. -R. Sun and M. -F. Wu, "Spontaneous Pair Production in Reissner-Nordstrom Black Holes," Phys. Rev. D 85, 124041 (2012) [arXiv:1202.3224 [hep-th]].
[9] V. Dzhunushaliev, "Canonical Conjugated Dirac Equation in a Curved Space," arXiv:1202.5100 [gr-qc].
[10] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, "Holographic Non-Fermi Liquid Fixed Points," Phil. Trans. Roy. Soc. A 369, 1640 (2011) [arXiv:1101.0597 [hep-th]].
[11] S. M. Carroll, "Spacetime and geometry: An introduction to general relativity," San Francisco, USA: Addison-Wesley (2004).
[12] N. J. Poplawski, "Spacetime and Fields," arXiv:0911.0334 [gr-qc].
[13] Wolfram MathWorld: The Web′s Most Extensive Mathematics Resource http://mathworld.wolfram.com/
指導教授 陳江梅(Chiang-mei Chen) 審核日期 2014-7-18
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